Dr. Andrew Nevai - Research Interests
Dr. Andrew Nevai
Research Interests (return to main page)

⛁ Research Summary
      ⛁ Modeling philosophy
⛁ Competition Between Species
      ⛁ Plant competition for sunlight
      ⛁ Species interactions at multiple scales (and spatial moment equations)
      ⛁ Competition and directed movement along resource gradients
⛁ Spatial Spread of Infectious Diseases
      ⛁ Continuous-time and discrete space (patch model)
      ⛁ Continuous-time and continuous-space (reaction-diffusion model)
      ⛁ Discrete-time and discrete-space (patch model)
⛁ Evolution of Optimal Choice
      ⛁ State-based decision-making in gray-jays
      ⛁ The honeybee nest-site selection process
      ⛁ Animal-mediated seed dispersal
⛁ Outlook

Research Summary (↑ top ↑)
I use mathematical approaches to address problems that arise in biology. Recently, my work has centered on three main areas of theoretical ecology: competition between species (plant competition for sunlight, interactions at multiple spatial scales, directed movement along resource gradients, and structured resources), the spatial spread of epidemic diseases (rabies and others), and the evolution of optimal choice (state-based decision-making in foraging gray jays, the honeybee nest-site selection process, and animal-mediated seed dispersal). I also am interested in species persistence and permanence within ecological communities; the dynamics of spatially (or otherwise) structured populations; classical and social foraging theory; animal and plant behavior; and formulating ecological models that make use of mechanistic reasoning and principles.

Modeling philosophy (↑ top ↑)
An ecological model consists of assumptions and rules that govern the behavior of an idealized ecological system. Model formulation is the exciting process by which these assumptions are articulated, and this requires a solid understanding of the fundamental ecological processes that are of interest. Sometimes, the simplest version of a model can be considered analytically, in which case its assumptions give rise to mathematical equations and ultimately to theorems that constitute its conclusions. Model formulation is also an iterative process, in which successive refinements arise as both analysis and intuition reveal departures from nature and ever better ways to capture biological realism. As a mathematical biologist, I must combine biological insight with appropriate analytical and computational methods to make the most of my mathematical models.

Competition Between Species (↑ top ↑)

Plant competition for sunlight (↑ top ↑)
I completed my doctoral dissertation on mathematical models of plant competition for sunlight under the co-direction of Paul Roberts (UCLA, Mathematics) and Richard Vance (UCLA, Ecology). We formulated and analyzed a canopy partitioning model (Vance and Nevai 2007, Nevai and Vance 2007, Nevai and Vance 2008) to determine whether two plant species with clonal growth forms and possessing distinct but overlapping vertical leaf profiles can successfully coexist while competing for sunlight and no other resource. The results of this base model are robust to a variety of density-independent and density-dependent enrichments.

I am collaborating with Winfried Just (Ohio University, Mathematics) on open questions posed in my Ph.D. dissertation. In (Just and Nevai 2008), we constructed an example in which two clonal plant species that obey a canopy partitioning model (Vance and Nevai 2007, Nevai and Vance 2007), and have rectangular vertical leaf profiles and distinct photosynthesis functions, can coexist stably at multiple equilibra. We also constructed a second example, in which the species share the same photosynthesis function but one species has a bi-rectangular vertical leaf profile. We recently submitted a second paper which explores plant competition for sunlight in which at least one species has a finitely supported vertical leaf profile (Just and Nevai in review).

Species interactions at multiple scales (and spatial moment equations) (↑ top ↑)
Ben Bolker (University of Florida, Biology) and I are studying connections between a stochastic spatial logistic equation with diffusion and its related spatial moment equations (Bolker and Pacala 1999, Dieckmann and Law 2000). I am also interested in connections between the fundamental growth equation of population ecology and its related spatial moment equations. Our results are currently in preparation.

Competition and directed movement along resource gradients (↑ top ↑)
Yuan Lou (Ohio State, Mathematics) and I are collaborating on a mathematical model that describes the interactions of spatially-structured populations. This continuous-time patch model will compare the outcome of competition between two Lotka-Volterra type species, both of which move randomly by diffusion but only one of which can also move intelligently toward patches with higher resource levels. It is expected that under some parameter combinations, coexistence will result in which the intelligent species concentrates in patches with high resource levels (relative to neighboring patches) and the other species subsists in the remaining patches. This work may confirm and extend the results of an earlier related partial differential equation competition model with ecological diffusion and intelligent movement along resource gradients (Lou 2005). This work will also evolve into other related projects.

Spatial Spread of Infectious Diseases (↑ top ↑)
Linda Allen (Texas Tech University, Mathematics and Statistics), Yuan Lou (Ohio State, Mathematics), Ben Bolker (University of Florida, Biology), and I are collaborating on a family of spatial epidemic disease models (SIS) in which the movement rate of one subpopulation (susceptible individuals) becomes very small relative to that of another (infected individuals). The project consists of mathematical analysis of a continuous-time discrete patch model (Allen et al. 2007), a continuous-time reaction-diffusion model (Allen et al. 2008), and a discrete-time patch model (to be submitted).

In each project, we connect spatial heterogeneity, habitat connectivity, and different movement rates among subpopulations to the observed spatial patterns of infectious diseases. Local differences in disease transmission and recovery rates characterize whether regioins are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. In low-risk habitats, the disease persists only when the mobility of infected individuals lies below some threshold value, but for high-risk habitats, the disease always persists. When the disease does persist, then there exists an endemic equilibrium (EE) which is unique and positive everywhere. This EE tends to a spatially inhomogeneous disease-free equilibrium (DFE) as the mobility of susceptible individuals tends to zero. Sufficient conditions for whether high-risk regions in the limiting DFE are empty can be given in terms of disease transmission and recovery rates, habitat connectivity, and the infected movement rate. For each model, we also compute the basic reproduction number as the spectral radius of the appropriate next generation operator.

In studying these models, we make use of comparison principles, the theory of nonnegative and irreducible matrices, the theory of elliptic operators, linear eigenvalue problems, the Perron-Frobenius Theorem for the eigenvalues of a nonnegative irreducible matrix, the Krein-Rutman Theorem for the eigenvalues of a positive linear operator, maximum principles for discrete and continuous systems, and super-solution and sub-solution methods for monotone dynamical systems.

Continuous-time disease patch model (↑ top ↑)
One of these models, which is in continuous-time and describes the movement of individuals between discrete patches, consists of a system of differential equations. More description to come soon...

Reaction-diffusion disease model (↑ top ↑)

Our continuous-space version of the model consists of a system of reaction diffusion equations. Several notable feature of this model are that the basic reproduction number is defined as the solution to a variational problem and the limiting DFE is defined as the solution to a free-boundary problem.

Discrete-time disease patch model (↑ top ↑)

Our discrete-time patch model consists of a system of difference equations. More description to come soon...

Evolution of Optimal Choice (↑ top ↑)

State-based decision-making in gray-jays (↑ top ↑)

Tom Waite (Ohio State, Evolution, Ecology, and Organismal Biology), Kevin Passino (Ohio State, Electrical Engineering), and I collaborated on a state-based individual foraging model for gray jays (Perisoreus canadensis). This model, which is based on a previous optimal choice model (Houston and McNamara 1999), incorporates partial preferencing to describe a tradeoff between maximizing reproductive value and minimizing predation risk for an individual that hoards tens of thousands of food items over the course of a year. This model will ultimately lead to generalized mathematical models of optimal choice that can be applied to a wide variety of individually foraging animal species. Two papers on this subject have now been published (Waite et al. 2007, Nevai et al. 2007).

The honeybee nest-site selection process (↑ top ↑)

I am collaborating with Kevin Passino (Ohio State, Electrical Engineering) on a mathematical model to describe the distributed self-organizing quorum-sensing decision-making process called nest-site selection in honeybees (Apis mellifera). We are formulating this model in continuous-time using a series of systems of ordinary differential equations in which new sites are discovered and considered asynchronously. This model is based on social insect foraging models and several previous nest-site selection models, including a continuous-time epidemiological-based SIR model (Britton et al. 2002), a discrete-time Leslie matrix model (Myerscough 2003) and a recent stochastic discrete-time model (Passino and Seeley 2005). Our model will demonstrate that natural selection has tuned certain parameters in this process so that the speed of decision-making (measured on the order of hours) is balanced against accuracy (so that a high quality nest-site is actually chosen). Our results are currently in preparation.

Animal-mediated seed dispersal (↑ top ↑)

Graduate student Vishwesha Guttal (Ohio State, Physics), Gregg Hartvigsen (SUNY Geneso, Biology), and I are examining the role of animal-mediated seed dispersal mechanisms (such as endozoochory and epizoochory) and animal movement behaviors (uncorrelated random walks, correlated random walks, and Levy flights) on the ability of plant species to obtain leptokurtic dispersal kernels. Our results are currently in preparation.

Outlook (↑ top ↑)
I find it natural to describe ecological problems using the language of mathematics because ecological communities are dynamical systems and evolution by natural selection favors optimization. The use of mathematics not only improves our understanding of ecology, but also offers new and interesting challenges to mathematicians. When mathematicians collaborate with biologists in problem solving, the resulting work can be realistic and useful to the biological community, it can deepen our understanding in the originating field, it can make creative use of both pure and applied mathematics, and it often leads to the development of new mathematical methods. I hope that my future research into mathematical ecology will allow me to find new connections between existing areas of mathematics and biology, and will also inspire new mathematics.